Theorem prnz 4729

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4714	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4293	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437  ∅c0 4282  {cpr 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-v 3439  df-dif 3901  df-un 3903  df-nul 4283  df-sn 4576  df-pr 4578

This theorem is referenced by:  opnz  5416  propssopi  5451  fiint  9218  wilthlem2  27007  upgrbi  29073  wlkvtxiedg  29605  shincli  31344  chincli  31442  constrextdg2lem  33782  spr0nelg  47600  sprvalpwn0  47607